Multifractal Analysis of Turbulence by Statistics based on Non-Extensive Tsallis' or Extensive R\'{e}nyi's Entropy

An analytical expression of probability density function (PDF) of velocity fluctuation is derived with the help of the statistics based on generalized entropy (the Tsallis entropy or the R'{e}nyi entropy). It is revealed that the derived PDF explains the detailed structure of experimentally observed PDF as well as the scaling exponents of velocity structure function. Every parameters appeared in the analysis, including the index proper to the Tsallis entropy or the R'{e}nyi entropy, are determined, self-consistently, by making use of observed value of intermittency exponent. The experiments conducted by Lewis and Swinney (1999) are analyzed.